Accuracy of a Recent Regularized Nuclear Potential

[Image: see text] F. Gygi recently suggested an analytic, norm-conserving, regularized nuclear potential to enable all-electron plane-wave calculations [Gygi J. Chem. Theory Comput.2023, 19, 1300–1309.]36757291. This potential V(r) is determined by inverting the Schrödinger equation for the wave function Ansatz ϕ(r) = exp[−h(r)]/√π with h(r) = r erf(ar) + b exp(−a(2)r(2)), where a and b are parameters. Gygi fixes b by demanding ϕ to be normalized, with the value b(a) depending on the strength of the regularization controlled by a. We begin this work by re-examining the determination of b(a) and find that the original 10-decimal tabulations of Gygi are only correct to 5 decimals, leading to normalization errors in the order of 10(–10). In contrast, we show that a simple 100-point radial quadrature scheme not only ensures at least 10 correct decimals of b but also leads to machine-precision level satisfaction of the normalization condition. Moreover, we extend Gygi’s plane-wave study by examining the accuracy of V(r) with high-precision finite element calculations with Hartree–Fock and LDA, GGA, and meta-GGA functionals on first- to fifth-period atoms. We find that although the convergence of the total energy appears slow in the regularization parameter a, orbital energies and shapes are indeed reproduced accurately by the regularized potential even with relatively small values of a, as compared to results obtained with a point nucleus. The accuracy of the potential is furthermore studied with s-d excitation energies of Sc–Cu as well as ionization potentials of He–Kr, which are found to converge to sub-meV precision with a = 4. The findings of this work are in full support of Gygi’s contribution, indicating that all-electron plane-wave calculations can be accurately performed with the regularized nuclear potential.